Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2013 / Differential Equations / 8 / a / Solution

Use the convention $A{u}_{xx}+2B{u}_{xy}+C{u}_{yy}=0$, with discriminant $B^2-AC$. Here $A=1$, $B=0$, $C=\mathrm{sign}(y)$.

For $y>0$, the discriminant is $-1<0$, so the equation is an elliptic partial differential equation, already in the canonical Laplace equation form $u_{xx}+u_{yy}=0$.

For $y<0$, the discriminant is $1>0$, so it is a hyperbolic partial differential equation, the wave equation $u_{xx}-u_{yy}=0$. Introduce characteristic coordinatesThen ${\partial }_x={\partial }_{\xi }+{\partial }_{\eta }$ and ${\partial }_y={\partial }_{\xi }-{\partial }_{\eta }$, givingOn $y=0$, the discriminant vanishes and the principal part has rank one; the degenerate parabolic type is $u_{xx}=0$. This is an interface line, rather than an open parabolic subdomain. ThusThis is a mixed-type partial differential equation. The classification concerns the principal part and remains meaningful even though the coefficient jumps at the interface.